Abundant new lump type interaction wave solutions and nonlinear dynamics for the (4+1)-dimensional Fokas equation

Junjie Li; Pei Lin; Xin Zhen; Junjiang Zhong; Junjie Li; Pei Lin; Xin Zhen; Junjiang Zhong

doi:10.3934/nhm.2026036

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 865-893. doi: 10.3934/nhm.2026036

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Research article

Abundant new lump type interaction wave solutions and nonlinear dynamics for the (4+1)-dimensional Fokas equation

1.
School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, China
2.
College of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China

Received: 24 November 2025 Revised: 24 April 2026 Accepted: 28 April 2026 Published: 07 May 2026

In this paper, the (4+1)-dimensional Fokas equation is first reduced to a (2+1)-dimensional bilinear form, and then analyzed by an architecture-guided bilinear neural network method (BNNM) within the Hirota framework. Three representative network topologies, namely the "3-2-1", "3-3-1", and "3-2-3-1" configurations, are used to organize trial functions and derive exact solutions through symbolic coefficient matching. As a result, several families of exact wave patterns are obtained, including lump solutions, lump–stripe-type solutions, lump–soliton-type solutions, and selected degenerate/high-amplitude structures. The comparison among different architectures shows that the hidden-layer depth and the activation composition affect the admissible ansatz forms and the morphology of the resulting interaction profiles for the reduced Fokas equation. The derived solutions are interpreted as prototype wave patterns and analytical benchmarks for localization, interaction, and modulation in multidimensional dispersive media, rather than as experimentally calibrated predictions. In addition, an auxiliary Duffing-based modulation diagnostic is used to visualize the irregular modulation of one selected high-amplitude profile; this comparison is qualitative and does not constitute a rigorous proof that the reduced Fokas equation itself is chaotic. These results show that the BNNM framework provides a structured symbolic route to construct and compare the exact interaction solutions of the reduced Fokas equation.
- Fokas equation,
- neural network,
- analytical solutions,
- Hirota bilinear method
Citation: Junjie Li, Pei Lin, Xin Zhen, Junjiang Zhong. Abundant new lump type interaction wave solutions and nonlinear dynamics for the (4+1)-dimensional Fokas equation[J]. Networks and Heterogeneous Media, 2026, 21(3): 865-893. doi: 10.3934/nhm.2026036

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Abstract

In this paper, the (4+1)-dimensional Fokas equation is first reduced to a (2+1)-dimensional bilinear form, and then analyzed by an architecture-guided bilinear neural network method (BNNM) within the Hirota framework. Three representative network topologies, namely the "3-2-1", "3-3-1", and "3-2-3-1" configurations, are used to organize trial functions and derive exact solutions through symbolic coefficient matching. As a result, several families of exact wave patterns are obtained, including lump solutions, lump–stripe-type solutions, lump–soliton-type solutions, and selected degenerate/high-amplitude structures. The comparison among different architectures shows that the hidden-layer depth and the activation composition affect the admissible ansatz forms and the morphology of the resulting interaction profiles for the reduced Fokas equation. The derived solutions are interpreted as prototype wave patterns and analytical benchmarks for localization, interaction, and modulation in multidimensional dispersive media, rather than as experimentally calibrated predictions. In addition, an auxiliary Duffing-based modulation diagnostic is used to visualize the irregular modulation of one selected high-amplitude profile; this comparison is qualitative and does not constitute a rigorous proof that the reduced Fokas equation itself is chaotic. These results show that the BNNM framework provides a structured symbolic route to construct and compare the exact interaction solutions of the reduced Fokas equation.

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